My thoughts on LG and JTE's crying scenes (note: as far as I was concerned, these crying scenes were natural and expected. I would have been disappointed with the depiction of the characters had they not happened):
They are deeply in love but cannot be together. JTE has to forsake her world to be with him, but she cannot leave her dad and friends in Korea, especially with a psycho and his army on the loose. LG once asked JTE why she became a cop and she said she decided to be brave because not everyone can be. There is only one way they can stop the freezing - close the portal between the worlds. If the portal is closed, she will be in Korea and he, in Corea. If they don't close the portal, time in both worlds will freeze. This will also cause them to be separated forever, or only to see each other in "frozen moments in time" as what happened in the bamboo forest when he was not "fully there" and in JTE's yard when LG brought her flowers and JTE sensed he was in another time. In addition to these, they are both in danger, either or both of them might not survive the clash with Lee Lim. Every time they meet, their happiness at seeing each other again as well as the fear it would be the last time they would see each other make them highly emotional.
Her crying scene in episodes 11 and 12 was due to above reasons plus the following: near-death experience (kidnapped, escaped from her kidnappers, almost run over by a massive truck, attacked by the driver, faced off an army of henchmen), exhaustion due to this experience as well as walking for who knows how many miles, dehydration, relief and pure bliss at seeing him again. An ordinary girl would not have survived this ordeal. Lee Gon's future queen just cried.